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I've been approved for several Amex business cards with absolutely no current business (just a theorized hypothetical plan of mentioning that I'm considering starting up consulting on the side as a business, if they would ever ask).

I simply applied giving my name as the business name, the minimum number of employees, $0 business income, 0 years in business, expected spending per month being the minimum spend (for that card's signup bonus) divided by the number of months to do that spend (so $1000 for a card that needs $3000 spend in the first 3 months). Approved every time with no questions asked, not even a conversation with anyone.

And I'm far from the only one.

So if people can get approved with no business at all, I don't see why you shouldn't be able to get approved with a real business but with no income that you can prove. Just say your business income is $0 and be done with it. Amex will care about your personal income, not your business income.